Multidimensional cosmological model describing the evolution of (n + 1) Einstein spaces in the theory with several scalar fields and forms is considered. When a (electro-magnetic composite) p-brane Ansatz is adopted the field equations are reduced to the equations for Toda-like system. The Wheeler-De Witt equation is obtained. In the case when n "internal" spaces are Ricci-flat, one space M 0 has a non-zero curvature, and all p-branes do not "live" in M 0 , the classical and quantum solutions are obtained if certain orthogonality relations on parameters are imposed. Spherically-symmetric solutions with intersecting non-extremal p-branes are singled out. A non-orthogonal generalization of intersection rules corresponding to (open, closed) Toda lattices is obtained. A chain of bosonic D ≥ 11 models (that may be related to hypothetical higher dimensional supergravities and F -theories) is suggested.

We use the general solution to the trace of the 4-dimensional Einstein equations for static, spherically symmetric configurations as a basis for finding a general class of black hole (BH) metrics, containing one arbitrary function gtt = A(r) which vanishes at some r = r h > 0, the horizon radius. Under certain reasonable restrictions, BH metrics are found with or without matter and, depending on the boundary conditions, can be asymptotically flat or have any other prescribed asymptotic. It is shown that our procedure generically leads to families of globally regular BHs with a Kerr-like global structure as well as symmetric wormholes. Horizons in space-times with zero scalar curvature are shown to be either simple or double. The same is generically true for horizons inside a matter distribution, but in special cases there can be horizons of any order. A few simple examples are discussed. A natural application of the above results is the brane world concept, in which the trace of the 4D gravity equations is the only unambiguous equation for the 4D metric, and its solutions can be continued into the 5D bulk according to the embedding theorems.Maeda and Sasaki [11] from 5-dimensional gravity with the aid of the Gauss and Codazzi equations [28]:

This short review deals with a multidimensional gravitational model containing dilatonic scalar fields and antisymmetric forms. The manifold is chosen in the product form. The sigma-model approach and exact solutions are reviewed. 1

A multidimensional gravitational model containing several dilatonic scalar fields and antisymmetric forms is considered. The manifold is chosen in the form M = M 0 × M 1 × . . . × M n , where M i are Einstein spaces (i ≥ 1). The block-diagonal metric is chosen and all fields and scale factors of the metric are functions on M 0 . For the forms composite (electro-magnetic) p-brane ansatz is adopted. The model is reduced to gravitating self-interacting sigma-model with certain constraints. In pure electric and magnetic cases the number of these constraints is n 1 (n 1 − 1)/2 where n 1 is number of 1-dimensional manifolds among M i . In the "electro-magnetic" case for dimM 0 = 1, 3 additional n 1 constraints appear. A family of "MajumdarPapapetrou type" solutions governed by a set of harmonic functions is obtained, when all factor-spaces M ν are Ricci-flat. These solutions are generalized to the case of non-Ricci-flat M 0 when also some additional "internal" Einstein spaces of non-zero curvature are added to M . As an example exact solutions for D = 11 supergravity and related 12-dimensional theory are presented.

We give a comparative description of different types of regular static, spherically symmetric black holes (BHs) and discuss in more detail their particular type, which we suggest to call black universes. The latter have a Schwarzschild-like causal structure, but inside the horizon there is an expanding Kantowski-Sachs universe and a de Sitter infinity instead of a singularity. Thus a hypothetic BH explorer gets a chance to survive. Solutions of this kind are naturally obtained if one considers static, spherically symmetric distributions of various (but not all) kinds of phantom matter whose existence is favoured by cosmological observations. It also looks possible that our Universe has originated from phantom-dominated collapse in another universe and underwent isotropization after crossing the horizon. An explicit example of a black-universe solution with positive Schwarzschild mass is discussed.

Multidimensional model describing the cosmological evolution of n Einstein spaces in the theory with l scalar fields and forms is considered. When electromagnetic composite p-brane ansatz is adopted, and certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity is reduced to a billiard on the (N − 1)-dimensional Lobachevsky space H N −1 , N = n + l. The geometrical criterion for the finiteness of the billiard volume and its compactness is used. This criterion reduces the problem to the problem of illumination of (N − 2)-dimensional sphere S N −2 by point-like sources. Some examples with billiards of finite volume and hence oscillating behaviour near the singularity are considered. Among them examples with square and triangle 2-dimensional billiards (e.g. that of the Bianchi-IX model) and a 4-dimensional billiard in "truncated" D = 11 supergravity model (without the Chern-Simons term) are considered. It is shown that the inclusion of the Chern-Simons term destroys the confining of a billiard.

The multidimensional cosmological model describing the evolution of n Einstein spaces is considered in the presence of a multicomponent perfect fluid. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity are reduced to a billiard on the (n-1)-dimensional Lobachevsky space . The geometrical criterion for the finiteness of the billiard volume and its compactness is suggested. This criterion reduces the problem to the problem of illumination of an (n-2)-dimensional sphere by point-like sources. Some generalizations of the considered scheme (including scalar field and quantum generalizations) are considered.

Black hole generalized p -brane solutions for a wide class of intersection rules are obtained. The solutions are defined on a manifold that contains a product of n − 1 Ricci-flat internal spaces. They are defined up to a set of functions H s obeying non-linear differential equations equivalent to Toda-type equations with certain boundary conditions imposed. A conjecture on polynomial structure of governing functions H s for intersections related to semisimple Lie algebras is suggested. This conjecture is proved for Lie algebras: A m , C m+1 , m ≥ 1 . For simple Lie algebras the powers of polynomials coincide with the components of twice the dual Weyl vector in the basis of simple coroots. The coefficients of polynomials depend upon the extremality parameter µ > 0 . In the extremal case µ = 0 such polynomials were considered previously by H. Lü, J. Maharana, S. Mukherji and C.N. Pope. Explicit formulas for A 2 -solution are obtained. Two examples of A 2 -dyon solutions, i.e. dyon in D = 11 supergravity with M 2 and M 5 branes intersecting at a point and Kaluza-Klein dyon, are considered.